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                <time class="level-item has-text-grey" datetime="2018-11-30T15:54:06.000Z">2018-11-30</time>
                
                
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                Learning notes_2018_12
            

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            <h2 id="😀-一、梯度、散度、旋度"><a href="#😀-一、梯度、散度、旋度" class="headerlink" title="😀 一、梯度、散度、旋度"></a>😀 一、梯度、散度、旋度</h2><p><a href="http://www.365yg.com/i6522440389700354564/#mid=1569021639000065" target="_blank" rel="noopener">轻松理解散度和旋度 - 数学知识的动画解析</a></p>
<h2 id="🖐-二、拉格朗日乘子法和KKT条件"><a href="#🖐-二、拉格朗日乘子法和KKT条件" class="headerlink" title="🖐 二、拉格朗日乘子法和KKT条件"></a>🖐 二、拉格朗日乘子法和KKT条件</h2><p><a href="http://slidesplayer.com/slide/11358544/22440389700354564/#mid=1569021639000065" target="_blank" rel="noopener">汽车优化设计 第二章：优化方法的数学基础 王琥 湖南大学 机械与运载工程学院</a></p>
<a id="more"></a> 
<p><img src="/2018/11/30/Learning-notes-2018-12/chrome_2018-12-04_20-22-17.png" alt></p>
<p><a href="https://zhuanlan.zhihu.com/p/26514613" target="_blank" rel="noopener">浅谈最优化问题的KKT条件</a></p>
<p><img src="/2018/11/30/Learning-notes-2018-12/chrome_2018-12-04_20-20-46.png" alt></p>
<p><a href="https://zhuanlan.zhihu.com/p/28804123" target="_blank" rel="noopener">【分类战车SVM】第四话：拉格朗日对偶问题（原来这么简单，你也可以轻松学会）</a></p>
<h2 id="🙎-三、点乘和叉乘"><a href="#🙎-三、点乘和叉乘" class="headerlink" title="🙎 三、点乘和叉乘"></a>🙎 三、点乘和叉乘</h2><script type="math/tex; mode=display">a\cdot b =|a||b|\cos \theta</script><script type="math/tex; mode=display">a\cdot b =a_1b_1+a_2b_2+ ... +a_nb_n</script><script type="math/tex; mode=display">a \times b= 
\left|
 \begin{matrix}
   i & j & k \\
   x_1 & y_1 & z_1  \\
   x_2 & y_2 & z_2  
  \end{matrix} 
\right|=(y_1z_2-y_2z_1)i-(x_1z_2-x_2z_1)j+(x_1y_2+x_2y_1)k</script><p><img src="/2018/11/30/Learning-notes-2018-12/timg.jpg" alt="点乘和叉乘"></p>
<h2 id="🧚‍-四、凸集、凸函数"><a href="#🧚‍-四、凸集、凸函数" class="headerlink" title="🧚‍ 四、凸集、凸函数"></a>🧚‍ 四、凸集、凸函数</h2><ol>
<li>凸集</li>
</ol>
<p>若某集合中的x和y两个点，若x和y连线之间的所有点（即0&lt;=μ&lt;=1，μx+(1-μ)y）仍属于这个集合，则称此集合为凸集。</p>
<p>直观几何表示：<br><img src="/2018/11/30/Learning-notes-2018-12/20141103170550131.jpg" alt="凸集和凹集"></p>
<p>左边的是凸集，右边的不是凸集，因为右边的集合中任意两点x和y连线之间的所有点有时不属于这个集合（右图中的连线）。</p>
<ol>
<li>凸函数</li>
</ol>
<p>对于$x$是定义在某凸集（非空的，空集也被规定为凸集）上的函数，对于凸集中的任意两点$x_1$和$x_2$，若</p>
<script type="math/tex; mode=display">f[\mu x_1+(1-\mu)x_2]\leq \mu f(x_1)+(1-\mu)f(x_2)</script><p>则称函数$f(x)$为<code>凸函数</code>。</p>
<p>直观几何表示：<br><img src="/2018/11/30/Learning-notes-2018-12/20141103171408380.jpg" alt="凸函数"></p>
<p>左边的是凸集，右边的不是凸集，因为右边的集合中任意两点x和y连线之间的所有点有时不属于这个集合（右图中的连线）。</p>
<p>也就是说两点对应的函数值$f(x_1)$和$f(x_2)$的之间的连线$(\mu f(x_1)+(1-\mu)f(x_2))$大于等于相应的（即同一个$\mu$值）两点之间连线$(\mu f(x_1)+(1-\mu)f(x_2))$对应的函数值$f[\mu x_1+(1-\mu)x_2]$</p>
<p>这其实应叫 下凸。</p>
<p>判定方法可利用<strong>定义法、已知结论法以及函数的二阶导数</strong></p>
<p>对于实数集上的凸函数，一般的判别方法是求它的二阶导数，如果其二阶导数在区间上非负，就称为凸函数。(向下凸)</p>
<p>如果其二阶导数在区间上恒大于0，就称为严格凸函数。</p>
<h2 id="👨‍-五、方程的解"><a href="#👨‍-五、方程的解" class="headerlink" title="👨‍ 五、方程的解"></a>👨‍ 五、方程的解</h2><p><img src="/2018/11/30/Learning-notes-2018-12/20141128173356875.jpg" alt="方程的解"><br>若矩阵A的秩为r，即r=R(A) ：</p>
<p>1）若$r=M=N$，则线性方程组$Ax=b$有唯一解；</p>
<p>2）若$r=N&lt; M$,则线性方程组$Ax=b$有唯一解或无解；</p>
<p>3）若$r=M&lt; N$，则线性方程组$Ax=b$有无穷解；</p>
<p>4）若$r&lt; M$,$r&lt; N$，则线性方程组$Ax =b$有无穷解或无解；</p>
<h2 id="🍳-六、正交投影和投影矩阵"><a href="#🍳-六、正交投影和投影矩阵" class="headerlink" title="🍳 六、正交投影和投影矩阵"></a>🍳 六、正交投影和投影矩阵</h2><p><img src="/2018/11/30/Learning-notes-2018-12/20141125122541339.jpg" alt="正交投影"></p>
<p><strong>正交投影</strong>$p$是向量$b$在平面（由矩阵$A$的列向量$a_1$和$a_2$确定）上正交投影</p>
<p>投影矩阵是从向量b变换到其正交投影p过程中的变换矩阵P：<br><img src="/2018/11/30/Learning-notes-2018-12/20141125123056796.jpg" alt="正交投影和投影矩阵"></p>
<h2 id="🍻-七、MP和OMP"><a href="#🍻-七、MP和OMP" class="headerlink" title="🍻 七、MP和OMP"></a>🍻 七、MP和OMP</h2><p><a href="https://korediantousman.staff.telkomuniversity.ac.id/files/2017/08/main.pdf" target="_blank" rel="noopener">MP</a><br>OMP</p>
<hr>
<p>以上。(づ●─●)づ</p>

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                    <a class="has-link-grey -link" href="/tags/KKT/">KKT</a>, <a class="has-link-grey -link" href="/tags/凸函数，正交和投影/">凸函数，正交和投影</a>, <a class="has-link-grey -link" href="/tags/拉格朗日乘子/">拉格朗日乘子</a>, <a class="has-link-grey -link" href="/tags/梯度散度和旋度/">梯度散度和旋度</a>
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        <a class="is-flex" href="#😀-一、梯度、散度、旋度">
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        <span>😀 一、梯度、散度、旋度</span>
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        <a class="is-flex" href="#🖐-二、拉格朗日乘子法和KKT条件">
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        <span>🖐 二、拉格朗日乘子法和KKT条件</span>
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